Representations for weighted Moore-Penrose inverses of partitioned adjointable operators

TL;DR

This paper investigates properties and representations of weighted Moore-Penrose inverses for partitioned adjointable operators on Hilbert C*-modules, extending known matrix results to a more general operator setting.

Contribution

It provides new unified and block-wise representations of weighted Moore-Penrose inverses for partitioned operators on Hilbert C*-modules, generalizing matrix results.

Findings

Unified representation for 1x2 partitioned case

Construction method from non-weighted to weighted inverses for 2x2 case

Generalization of matrix results to Hilbert C*-module operators

Abstract

For two positive definite adjointable operators $M$ and $N$, and an adjointable operator $A$ acting on a Hilbert $C^*$-module, some properties of the weighted Moore-Penrose inverse $A^\dag_{MN}$ are established. When $A=(A_{ij})$ is $1\times 2$ or $2\times 2$ partitioned, general representations for $A^\dag_{MN}$ in terms of the individual blocks $A_{ij}$ are studied. In case $A$ is $1\times 2$ partitioned, a unified representation for $A^\dag_{MN}$ is presented. In the $2\times 2$ partitioned case, an approach to constructing Moore-Penrose inverses from the non-weighted case to the weighted case is provided. Some results known for matrices are generalized in the general setting of Hilbert $C^*$-module operators.